id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-daf50fc2e26055f099c4e5d2120ecec99c4925d1019226919aaecd10c8ac26cf | A computer generates pairs $(x, y)$ with $x, y \in (0,1)$ and computes $a = x + ym$ and $b = y + xm$, where $m$ is a positive integer. The process continues until all pairs $(a, b)$ with integer values of $a$ and $b$ are determined. The computer takes 5 seconds to determine each such pair. If the computer needed 595 se... | [
"11"
] | [
"A computer generates pairs $(x, y)$ with $x, y \\in (0,1)$ and computes $a = x + ym$ and $b = y + xm$, where $m$ is a positive integer. The process continues until all pairs $(a, b)$ with integer values of $a$ and $b$ are determined. The computer takes 5 seconds to determine each such pair. If the computer needed ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-9e6b1db5fa8662ceee709217463dc6652cc434216ca3cd9964bcc1d6ebd59fcf | A computer is programmed to randomly generate a string of six symbols using only the letters A, B, and C. What is the probability that the string will not contain three consecutive A's? | [
"\\(\\frac{8}{9}\\)"
] | [
"A computer is programmed to randomly generate a string of six symbols using only the letters A, B, and C. What is the probability that the string will not contain three consecutive A's?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 33
} | 48 | 0 | 0 |
omr-ws-84598103392fc9bbaf344b94046ece3a87895fe8dae409f8f68366613bfd27cc | A computer operator can enter 75 data forms each hour. After the operator has been working for 20 minutes, a second operator starts working also. Two hours later, they have entered 295 forms. How long would it have taken the first operator to enter 295 forms alone? | [
"\\dfrac{59}{15}"
] | [
"A computer operator can enter 75 data forms each hour. After the operator has been working for 20 minutes, a second operator starts working also. Two hours later, they have entered 295 forms. How long would it have taken the first operator to enter 295 forms alone?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12366346204168614a10cabab2fc063f18cb96fb603a94ce2bfd3840d1237cc6 | A computer prints out one number from 1 to 5 inclusive with equal likelihood every 15 minutes. What is the probability that, after 1 hour, the product of the numbers printed out is divisible by 10? | [
"\\(\\frac{304}{625}\\)"
] | [
"A computer prints out one number from 1 to 5 inclusive with equal likelihood every 15 minutes. What is the probability that, after 1 hour, the product of the numbers printed out is divisible by 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-fd217143a398f3bf6203cb42e7ef5dea6275195f61e8b9c7692e2a0abb4a13a7 | A computer program called $DuDu$ generates numbers $1, 2$ or $3$ at random once every one minute. It is known that the probability of getting the numbers $1, 2, 3$ in a row is the same, which is $\frac{1}{3}$. If $DuDu$ is considered to be turned on at minute $0$, and the probability that the sum of all numbers appeari... | [
"4"
] | [
"A computer program called $DuDu$ generates numbers $1, 2$ or $3$ at random once every one minute. It is known that the probability of getting the numbers $1, 2, 3$ in a row is the same, which is $\\frac{1}{3}$. If $DuDu$ is considered to be turned on at minute $0$, and the probability that the sum of all numbers a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ded9116417aa20c0029d2c435bbb5f78986fa56a0cc8976d503204d8027b0e4 | A computer program consists of two blocks written independently by two different programmers. The first block has an error with probability 0.2, and the second block has an error with probability 0.3. If the program returns an error, what is the probability that there is an error in both blocks? | [
"\\(\\frac{3}{22}\\)"
] | [
"A computer program consists of two blocks written independently by two different programmers. The first block has an error with probability 0.2, and the second block has an error with probability 0.3. If the program returns an error, what is the probability that there is an error in both blocks?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-50625cf479e0ae5a7a7e76d02faa873c5a574ddaa52ec9031e72f8a1b846e7a7 | A computer program generated 175 positive integers at random, none of which had a prime divisor greater than 10. Let \( N \) be the number of triples of these integers whose product is the cube of an integer. Calculate the minimum possible value of \( N \). | [
"66151"
] | [
"A computer program generated 175 positive integers at random, none of which had a prime divisor greater than 10. Let \\( N \\) be the number of triples of these integers whose product is the cube of an integer. Calculate the minimum possible value of \\( N \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3d6ee29bb72dec1e119373f2366e12fab3fbd211f3f2b429465da3e1b8f1bb6 | A computer program prints out all integers from 0 to ten thousand in base 6. How many numerals would it have printed? | [
"3742"
] | [
"A computer program prints out all integers from 0 to ten thousand in base 6. How many numerals would it have printed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7eea9ef8d9abd5980f3227f005d0ee5d80cb7a9d623f7a4001ee508bff531509 | A computer programmer wants to type all integers from 1 to 10000 in ascending order but omits numbers containing the digits 2 and 5. What number will he type in the 2014th place? | [
"3736"
] | [
"A computer programmer wants to type all integers from 1 to 10000 in ascending order but omits numbers containing the digits 2 and 5. What number will he type in the 2014th place?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-deaaccef27ffaf4e2237026138eb7d7fb4bd5a6582389aa128c674f86ee30a0f | A computer randomly assigns labels from $1$ to $64$ to an $8 \times 8$ electronic board twice. Let $n_k$ be the label of the square that was originally assigned $k$. Given that $n_{17} = 18$, find the probability that the sum $|n_1-1| + |n_2-2| + |n_3-3| + \cdots + |n_{64}-64| = 2018$. | [
"0"
] | [
"A computer randomly assigns labels from $1$ to $64$ to an $8 \\times 8$ electronic board twice. Let $n_k$ be the label of the square that was originally assigned $k$. Given that $n_{17} = 18$, find the probability that the sum $|n_1-1| + |n_2-2| + |n_3-3| + \\cdots + |n_{64}-64| = 2018$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-705e7988feab87548e6de31d669160eb50cf3d78e1433e144f02e0de2e394bca | A computer scans the digits of counting numbers. If it scans the first 1788 digits, what is the last counting number scanned? | [
"632"
] | [
"A computer scans the digits of counting numbers. If it scans the first 1788 digits, what is the last counting number scanned?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-24f22a3e00870efad070f63eebcc1b7c424c5b235eedf96bb8a974a028377e66 | A computer scientist needs to evaluate the expression \(\log \sum_{i=1}^{2021} e^{x_i}\) where \( x_i \) are real numbers. However, evaluating \( e^{x_i} \) for \( x_i > 88 \) causes the app to crash. Reformulate the expression so that it does not crash the app. | [
"\\log \\sum_{i=1}^{2021} e^{x_i} = \\max(x_i) + \\log \\sum_{i=1}^{2021} e^{x_i - \\max(x_i)}"
] | [
"A computer scientist needs to evaluate the expression \\(\\log \\sum_{i=1}^{2021} e^{x_i}\\) where \\( x_i \\) are real numbers. However, evaluating \\( e^{x_i} \\) for \\( x_i > 88 \\) causes the app to crash. Reformulate the expression so that it does not crash the app."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60b7489fe094ef911677a424a6c28403c9f57ced7d5229290c1adf07cd46f830 | A computer store sold a total of 300 items last month. The store sold six times as many hard drives as they did CD-ROM drives, and half as many floppy drives as hard drives. Based on this information, how many of each item were sold? | [
"90"
] | [
"A computer store sold a total of 300 items last month. The store sold six times as many hard drives as they did CD-ROM drives, and half as many floppy drives as hard drives. Based on this information, how many of each item were sold?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-256ba7c237e186bb4a772aa825e4ee7e23a4fe6da1799f6e2dfdafca08ae5d80 | A computer virus starts off infecting a single device. Every second, an infected computer has a $\frac{7}{30}$ chance to stay infected and not do anything else, a $\frac{7}{15}$ chance to infect a new computer, and a $\frac{1}{6}$ chance to infect two new computers. Otherwise (a $\frac{2}{15}$ chance), the virus gets e... | [
"\\(\\frac{4}{5}\\)"
] | [
"A computer virus starts off infecting a single device. Every second, an infected computer has a $\\frac{7}{30}$ chance to stay infected and not do anything else, a $\\frac{7}{15}$ chance to infect a new computer, and a $\\frac{1}{6}$ chance to infect two new computers. Otherwise (a $\\frac{2}{15}$ chance), the vir... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-a134ae0da92cf1c178be74688729b02c38e09ad207d11348915f0639bc845fac | A concert hall contains 400 seats. If \( N \) of these are sold at 3 s. 6 d. each and the rest at 5 s. each, find a formula for the total takings. | [
"\\( 24000 - 18N \\)"
] | [
"A concert hall contains 400 seats. If \\( N \\) of these are sold at 3 s. 6 d. each and the rest at 5 s. each, find a formula for the total takings."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-8630fa56cb4ee3d9571a394ea7f3b9c35746805cff72198e530150ec97cd00aa | A cone and a cylinder have a common base, and the vertex of the cone is in the center of the other base of the cylinder. Find the angle between the axis of the cone and its element if the ratio of the total surface area of the cylinder to the total surface area of the cone is $7: 4$. | [
"\\(\\alpha = \\arcsin\\frac{3}{5}\\)"
] | [
"A cone and a cylinder have a common base, and the vertex of the cone is in the center of the other base of the cylinder. Find the angle between the axis of the cone and its element if the ratio of the total surface area of the cylinder to the total surface area of the cone is $7: 4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cc7fbf7d39132a417fc2d889008ce4b34b1843c86f78d62156e87cf2f47c81c | A cone circumscribes a sphere with radius 1. Express the volume \( V \) of the cone as a function of the base radius \( r \). | [
"\\( V = \\frac{1}{3} \\pi (r^3 + r^2) \\)"
] | [
"A cone circumscribes a sphere with radius 1. Express the volume \\( V \\) of the cone as a function of the base radius \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1e237cdc830b658b55959556ab107e50f54d2cd12985636090a102a7ddeb921 | A cone has a base circumference of $3\pi$ cm and a circular cross section through point $B$ parallel to the base with a circumference of $\pi$ cm. The height of point $B$ above the base is $3\sqrt{7}$ cm. A path is drawn on the surface of the cone from point $A$ on the circumference of the base to point $B$ that wraps ... | [
"160 - 48\\sqrt{2}"
] | [
"A cone has a base circumference of $3\\pi$ cm and a circular cross section through point $B$ parallel to the base with a circumference of $\\pi$ cm. The height of point $B$ above the base is $3\\sqrt{7}$ cm. A path is drawn on the surface of the cone from point $A$ on the circumference of the base to point $B$ tha... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-1c770609d2c50dbbca5f453d660f545a9dc5b8124a02765eb560b9efa29c5058 | A cone has a base diameter of 1 cm and a slant height of 3 cm. From a point A halfway up the side of the cone, a string is passed twice around the cone to a point B on the circumference of the base. Find the length of the string. | [
"\\(\\frac{3 \\sqrt{7}}{2}\\)"
] | [
"A cone has a base diameter of 1 cm and a slant height of 3 cm. From a point A halfway up the side of the cone, a string is passed twice around the cone to a point B on the circumference of the base. Find the length of the string."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5d535b28fbc5844b6364380eace1e20d905f9afa333751f7effc570d1b029d6f | A cone has a base diameter of 1 unit and a slant height of 3 units. From a point A halfway up the side of the cone, a string is passed twice around it to come to a point B on the circumference of the base, directly below A. The string is then pulled until taut. How far is it from A to B along this taut string? | [
"\\(\\frac{3}{2}\\sqrt{7}\\)"
] | [
"A cone has a base diameter of 1 unit and a slant height of 3 units. From a point A halfway up the side of the cone, a string is passed twice around it to come to a point B on the circumference of the base, directly below A. The string is then pulled until taut. How far is it from A to B along this taut string?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-9129914615fad6e7e840cdaa0846e0de3a77dd925b0cb95491e0b3eb51dd13a7 | A cone has a base diameter of 1 unit and a slant height of 3 units. From a point \( A \) halfway up the side of the cone, a string is passed twice around the cone to come to a point \( B \) on the circumference of the base, directly below \( A \). The string is then pulled until taut. What is the distance from \( A \) ... | [
"\\(\\frac{3\\sqrt{7}}{2}\\)"
] | [
"A cone has a base diameter of 1 unit and a slant height of 3 units. From a point \\( A \\) halfway up the side of the cone, a string is passed twice around the cone to come to a point \\( B \\) on the circumference of the base, directly below \\( A \\). The string is then pulled until taut. What is the distance fr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-11f60da9b671834af376e013237c999ba20a96a6d2da5151a4fb8803b49700db | A cone has a height of 24 cm and a radius of 12 cm. What is the maximum volume of a cylinder that can fit inside this cone? | [
"\\( 512\\pi \\)"
] | [
"A cone has a height of 24 cm and a radius of 12 cm. What is the maximum volume of a cylinder that can fit inside this cone?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 29
} | 52 | 0 | 0 |
omr-ws-b0e68bab2c01790d77e0151d47c77f48c508a92da23a16da1155810869acb23d | A cone has a height of 4 cm and the angle between the cone's generatrix and the base plane is 30 degrees. Calculate the area of the cross-section formed by two mutually perpendicular generatrices. | [
"\\(\\frac{32}{3}\\)"
] | [
"A cone has a height of 4 cm and the angle between the cone's generatrix and the base plane is 30 degrees. Calculate the area of the cross-section formed by two mutually perpendicular generatrices."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-917fd0ed2cc3686f574804364d5194b168a32d7571e0fdc955aab060a2f6b3b4 | A cone has a height of 8 feet and a radius of 3 feet. If the water inside the cone is 4 feet deep, what is the surface area of the water? | [
"\\dfrac{9}{4}\\pi"
] | [
"A cone has a height of 8 feet and a radius of 3 feet. If the water inside the cone is 4 feet deep, what is the surface area of the water?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ae809908a153489b9e7a8ca43512ed02291c8ed80dec640499e2fba42f40321 | A cone has a radius of 1 and a height of 6. A cube is placed inside the cone such that the bottom of the cube is level with the bottom of the cone. What is the volume of the cube? | [
"\\(\\left( \\frac{6\\sqrt{3}}{9+\\sqrt{3}} \\right)^3\\)"
] | [
"A cone has a radius of 1 and a height of 6. A cube is placed inside the cone such that the bottom of the cube is level with the bottom of the cone. What is the volume of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b98535703e355555386a694597432e3c9e381d2fd2eb65101284197077e81e3 | A cone has a radius of 3 and a height of 4. An infinite number of spheres are placed inside the cone such that sphere \( C_0 \) is tangent to the base and the curved surface of the cone at more than one point, and for \( i \geq 1 \), sphere \( C_i \) is externally tangent to sphere \( C_{i-1} \) and internally tangent ... | [
"39"
] | [
"A cone has a radius of 3 and a height of 4. An infinite number of spheres are placed inside the cone such that sphere \\( C_0 \\) is tangent to the base and the curved surface of the cone at more than one point, and for \\( i \\geq 1 \\), sphere \\( C_i \\) is externally tangent to sphere \\( C_{i-1} \\) and inter... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-e54292b7ab64c288fca6380183b315575950ade4ac34ca37bea7b2c080a82111 | A cone has a volume of $12288\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy] import markers; size(150); import geometry; draw(scale(1,.2)*arc((0,0),1,0,180),dashed); draw(scale(1,.2)*arc((0,0... | [
"48.0"
] | [
"A cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]\n\nimport markers;\nsize(150);\nimport geometry;\ndraw(scale(1,.2)*arc((0,0),1,0,180),dashed);\ndraw(scale(1... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14a82a18faa6b2afe79f51837b88c1a21a54b8b3caf9bec2303600e70c0d0257 | A cone has an angle of $30^\circ$ at its vertex, and a sphere with radius $1$ is inscribed within it. A total of $n$ spheres are inscribed within the cone such that each sphere after the first is tangent to the sphere drawn before it and to the cone, and each sphere is consecutively larger than the last. After $n$ sphe... | [
"\\(\\frac{\\pi}{3} \\left( -2\\sqrt{6} - 2\\sqrt{2} + 2 + (11\\sqrt{6} + 16\\sqrt{3} - 12\\sqrt{2} - 29) \\left( 1 + \\frac{2}{\\sqrt{6} + \\sqrt{2} - 1} \\right)^{3n} \\right)\\)"
] | [
"A cone has an angle of $30^\\circ$ at its vertex, and a sphere with radius $1$ is inscribed within it. A total of $n$ spheres are inscribed within the cone such that each sphere after the first is tangent to the sphere drawn before it and to the cone, and each sphere is consecutively larger than the last. After $n... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d57b6914b50b0f76c2f0c063aced307e07a38f64e08b968ce724c533462a7d91 | A cone is circumscribed about a regular hexagonal pyramid. Find its volume, if the lateral edge of the pyramid is equal to $l$ and the face angle between two adjacent lateral edges is equal to $\alpha $. | [
"$V=\\frac{8}{3}\\pi l^3\\sin^2 \\frac{\\alpha }{2} \\sqrt {\\sin\\left(30^\\circ+\\frac{\\alpha }{2}\\right)\\sin\\left(30^\\circ- \\frac{\\alpha }{2}\\right)}.$"
] | [
"A cone is circumscribed about a regular hexagonal pyramid. Find its volume, if the lateral edge of the pyramid is equal to $l$ and the face angle between two adjacent lateral edges is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82e7dad25ce37aead252f5006e739268f4bcac930f39737200040162e5030272 | A cone is circumscribed around a triangular regular prism such that the bottom base of the prism is included in the base of the cone and the top vertices of the prism touch the side surface of the cone. All edges of the prism are of the same length. Calculate the volume of this cone and the tangent of the dilation angl... | [
"\\dfrac{\\sqrt{3}}{3}"
] | [
"A cone is circumscribed around a triangular regular prism such that the bottom base of the prism is included in the base of the cone and the top vertices of the prism touch the side surface of the cone. All edges of the prism are of the same length. Calculate the volume of this cone and the tangent of the dilation... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-02ef9b2764918b921161f58f127996830775436cbbd796bab50caa780e2891e0 | A cone is constructed with a semicircular piece of paper with radius 10. Find the height of the cone. | [
"5\\sqrt{3}"
] | [
"A cone is constructed with a semicircular piece of paper with radius 10. Find the height of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f929e5fda496ef32acfe5027afe8d6a23d6c14bf0e85fb412be4eca9da8ed672 | A cone is divided by two planes drawn parallel to the base into three parts having equal volume. What is the ratio of the radius of the base of the three parts so formed? | [
"\\(\\sqrt[3]{3}:\\sqrt[3]{2}:1\\)"
] | [
"A cone is divided by two planes drawn parallel to the base into three parts having equal volume. What is the ratio of the radius of the base of the three parts so formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 10
} | 13 | 0 | 0 |
omr-ws-7ac28bf40d95a9dba4aa178dfeed3d618f7ed1d0389ed09ee8701bb2ab07d8b2 | A cone is formed by lines passing through the point $(0, 0, 1)$ and the circle $\{(x, y, 0) : (x - 1)^2 + y^2 = 1\}$. If a plane $z = \alpha x + \beta$ with $\alpha > 0$ and $\beta \ge 0$ intersects the cone in a circle, then what is $\alpha$? | [
"1"
] | [
"A cone is formed by lines passing through the point $(0, 0, 1)$ and the circle $\\{(x, y, 0) : (x - 1)^2 + y^2 = 1\\}$. If a plane $z = \\alpha x + \\beta$ with $\\alpha > 0$ and $\\beta \\ge 0$ intersects the cone in a circle, then what is $\\alpha$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-9cdbb9a18f92a7f9becaa573060e97da2a5695e5d9f3d60536ba9cc2d0932b4d | A cone is formed from a 300-degree sector of a circle of radius 18 by aligning the two straight sides. [asy] size(110); draw(Arc((0,0),1,0,300)); draw((1,0)--(0,0)--(.5,-.5*sqrt(3))); label("18",(.5,0),S); label("$300^\circ$",(0,0),NW); [/asy] What is the result when the volume of the cone is divided by $\pi$? | [
"225\\sqrt{11}"
] | [
"A cone is formed from a 300-degree sector of a circle of radius 18 by aligning the two straight sides. [asy]\nsize(110);\ndraw(Arc((0,0),1,0,300));\ndraw((1,0)--(0,0)--(.5,-.5*sqrt(3)));\nlabel(\"18\",(.5,0),S); label(\"$300^\\circ$\",(0,0),NW);\n[/asy] What is the result when the volume of the cone is divided by ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b3df3458d6631fd457ba856db807ac2ba38c783cca5081372c696394e308c4d | A cone is inscribed in a regular triangular pyramid. Find the volume of the cone if the lateral edge of the pyramid is equal to $l$ and the face angle between two adjacent lateral edges is equal to $\alpha $. | [
"$V=\\frac{\\pi l^3\\sin^2 \\frac{\\alpha }{2}}{9\\sqrt 3} \\sqrt {3-4\\sin^2 \\displaystyle\\frac{\\alpha }{2}}$ $=\\displaystyle\\frac{2\\pi l^3\\sin^2 \\displaystyle\\frac{\\alpha }{2}\\sqrt {\\cos\\left(30^\\circ+\\displaystyle\\frac{\\alpha }{2} \\right)\\cos\\left(30^\\circ-\\displaystyle\\frac{\\alpha }{2}\... | [
"A cone is inscribed in a regular triangular pyramid. Find the volume of the cone if the lateral edge of the pyramid is equal to $l$ and the face angle between two adjacent lateral edges is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22bdfef22a591e54730c30862ea3e2ed94315fffedad2d75c9c56f82fc4f77c9 | A cone is inscribed in a sphere and its volume is equal to one fourth of that of the sphere. Find the volume of the sphere if the altitude of the cone is equal to $H$. | [
"The problem has two solutions: $V=\\frac{4}{3}\\pi H^3$ and $V=\\frac{4}{3}\\pi \\left(\\sqrt 5-2\\right)H^3$"
] | [
"A cone is inscribed in a sphere and its volume is equal to one fourth of that of the sphere. Find the volume of the sphere if the altitude of the cone is equal to $H$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c95047c0172f37ba200ad14f19e195bc631f144a66a1956815d99fd394dedda | A cone is inscribed in a sphere of radius 2 such that its base is one unit from the bottom of the sphere. What is the volume of the cone? | [
"\\( 3\\pi \\)",
"\\(3\\pi\\)"
] | [
"A cone is inscribed in a sphere of radius 2 such that its base is one unit from the bottom of the sphere. What is the volume of the cone?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 54
} | 93 | 0 | 0 |
omr-ws-6dd5b124f23eea9066f2351a97c38eb54362a8c3bfbf308b125953cecccb4410 | A cone is inscribed in a sphere whose surface area is $S$. The angle between the generatrix of the cone and the plane of the base is $\alpha $. Find the area of the total surface of the cone. | [
"$S\\sin \\alpha \\sin 2\\alpha \\cos ^2 \\frac{\\alpha }{2}$"
] | [
"A cone is inscribed in a sphere whose surface area is $S$. The angle between the generatrix of the cone and the plane of the base is $\\alpha $. Find the area of the total surface of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b4a13096b684ada8a5ec5f9c68a1bc2f44f819b7af4a34f9cdad6d4f30f3161 | A cone is inscribed in a sphere. The area of the lateral surface of the cone is $\frac{3}{8}$ the area of the sphere. Find the angle of inclination of the generatrix of the cone to the base plane. | [
"\\(60^\\circ, \\arccos \\frac{\\sqrt{13}-1}{4}\\)"
] | [
"A cone is inscribed in a sphere. The area of the lateral surface of the cone is $\\frac{3}{8}$ the area of the sphere. Find the angle of inclination of the generatrix of the cone to the base plane."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 5,
"tir": 14
} | 60 | 0 | 0 |
omr-ws-5cc5846c8e2a14cc3107a5c3d1848d22cd3d5b9064b0866187b4b60b2b3d4048 | A cone is inverted and filled with water to 3/4 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth. (You should enter 10.0000 for $10\%$ instead of 0.1000.) | [
"42.1875"
] | [
"A cone is inverted and filled with water to 3/4 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth. (You should enter 10.0000 for $10\\%$ instead of 0.1000.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1fef81a58f9d8907968e2bd7a33df3cd4a0fed36633a6ecaf4708eb3c71dfc64 | A cone of maximum surface area is inscribed within a sphere of radius 10. What is the surface area of this cone? | [
"\\dfrac{25(107 + 51\\sqrt{17})}{32} \\pi"
] | [
"A cone of maximum surface area is inscribed within a sphere of radius 10. What is the surface area of this cone?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-4e2e55f1a174af3a72020ff2087a63be53eb2c6e646c604d7f674376def6fc77 | A cone of minimum volume is circumscribed about a sphere of radius 10. What is the volume of this cone? | [
"\\(\\frac{8000}{3}\\pi\\)"
] | [
"A cone of minimum volume is circumscribed about a sphere of radius 10. What is the volume of this cone?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-17117b43356226cb9fb00b065855fbe25db777d0566ca323edf1131ad127a69f | A cone of radius $4$ and height $7$ has $A$ as its apex and $B$ as the center of its base. A second cone of radius $3$ and height $7$ has $B$ as its apex and $A$ as the center of its base. What is the volume of the region contained in both cones? | [
"\\(\\frac{48\\pi}{7}\\)"
] | [
"A cone of radius $4$ and height $7$ has $A$ as its apex and $B$ as the center of its base. A second cone of radius $3$ and height $7$ has $B$ as its apex and $A$ as the center of its base. What is the volume of the region contained in both cones?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-a3e888a2499126cd8998f40dc67da4fd282af59f213d5e6132de671984495c87 | A cone with a base radius of 30 cm and a height of 40 cm is cut horizontally 7 times such that the remaining solids have equal volume, with the topmost solid still being a cone. Find the height of the resulting cone. | [
"20"
] | [
"A cone with a base radius of 30 cm and a height of 40 cm is cut horizontally 7 times such that the remaining solids have equal volume, with the topmost solid still being a cone. Find the height of the resulting cone."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-e706cdd6489a02ccc066bb39158311240ea47d2ad3b344326a2bdb5a6b3cb6c5 | A cone with a height of 24 inches has a base with a radius of 18 inches. A plane perpendicular to the height intersects the cone halfway between its apex and base. What is the surface area of the frustum? Express the answer in terms of $\pi$. | [
"810\\pi"
] | [
"A cone with a height of 24 inches has a base with a radius of 18 inches. A plane perpendicular to the height intersects the cone halfway between its apex and base. What is the surface area of the frustum? Express the answer in terms of $\\pi$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-4c2f7857e6f2326e2299c355bcdb5c580f99ed5b0d224059fc5fa19fbbeed842 | A cone with height $6$ and base radius $r$ points upward with its apex at $(0,4)$. A second cone, identical to the first and also pointing upward, has its apex at $(0,0)$. Let $U$ be the volume of the union of the two cones and $I$ be the volume of the intersection of the two cones. If $U-I=\frac{13\pi}{6}$, compute $r... | [
"\\dfrac{3}{4}"
] | [
"A cone with height $6$ and base radius $r$ points upward with its apex at $(0,4)$. A second cone, identical to the first and also pointing upward, has its apex at $(0,0)$. Let $U$ be the volume of the union of the two cones and $I$ be the volume of the intersection of the two cones. If $U-I=\\frac{13\\pi}{6}$, com... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-32c4df20b09e94bacaac1e9b9de6f0b2d274065ab0e36263d51556a6bfaf1c2c | A cone with height 12 ft and radius 4 ft, pointing downward, is filled with water to a depth of 9 ft. Find the work required to pump all the water out of the top. | [
"8845.2\\pi"
] | [
"A cone with height 12 ft and radius 4 ft, pointing downward, is filled with water to a depth of 9 ft. Find the work required to pump all the water out of the top."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-25396a1e5cdbf950717a235bac80510ef26239ba243262b56a6e3a5b548a3552 | A cone with height 4 and radius 3 is filled halfway by volume with kool aid. What is the cube of the height of the filled portion? | [
"8"
] | [
"A cone with height 4 and radius 3 is filled halfway by volume with kool aid. What is the cube of the height of the filled portion?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-507fe153f3946fd9e67ad9a6db3e45c8aa5b91952a07eb3d9bf48aceaf051622 | A cone with height \( h \) and base radius \( R \) is cut such that the top is a circle with radius \( r \). Find the volume of the remaining part of the cone expressed in terms of \( R \), \( r \), and \( h \). What would you expect when \( r \) gets closer and closer to \( R \), and what would you expect when \( r = ... | [
"\\(\\frac{\\pi h (R^2 + Rr + r^2)}{3}\\)"
] | [
"A cone with height \\( h \\) and base radius \\( R \\) is cut such that the top is a circle with radius \\( r \\). Find the volume of the remaining part of the cone expressed in terms of \\( R \\), \\( r \\), and \\( h \\). What would you expect when \\( r \\) gets closer and closer to \\( R \\), and what would yo... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 13
} | 16 | 0 | 0 |
omr-ws-76f93d4c3c1b4bd0d616d983b657c29a1a2e9b1b3e479cd7efc33a338e878830 | A cone with radius $2 \text{ cm}$ and height $5 \text{ cm}$ is lowered point first into a tall cylinder with radius $7 \text{ cm}$ that is partially filled with water. Determine how fast the depth of the water is changing at the moment when the cone is completely submerged, given that the cone is moving with a speed of... | [
"\\dfrac{12}{49}"
] | [
"A cone with radius $2 \\text{ cm}$ and height $5 \\text{ cm}$ is lowered point first into a tall cylinder with radius $7 \\text{ cm}$ that is partially filled with water. Determine how fast the depth of the water is changing at the moment when the cone is completely submerged, given that the cone is moving with a ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-6fce4c4a12cfd026df13e60cc917f6059bfb40b762ff3f564b9ff6e3db99d0d1 | A cone with radius \( r \) and lateral edge \( l \) is filled with water. A largest possible iron sphere is lowered into the cone so that no part of it is outside the liquid. What volume of water overflows? | [
"\\(\\frac{4}{3} \\pi \\left( \\frac{r \\sqrt{l^2 - r^2}}{\\sqrt{l^2 - r^2} + r} \\right)^3\\)"
] | [
"A cone with radius \\( r \\) and lateral edge \\( l \\) is filled with water. A largest possible iron sphere is lowered into the cone so that no part of it is outside the liquid. What volume of water overflows?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-6a4c1169bc3dee0f8bcc19c6b238806bd5e772a11a4df23e4a47d7d43f4221dc | A cone with radius \( r \) and lateral edge \( l \) is filled with water. The largest possible iron sphere is lowered into the cone so that no part of it is outside the liquid. What volume of water overflows? | [
"\\(\\frac{4}{3}\\pi \\frac{r^3}{(l+r)^3}(l^2-r^2)^{\\frac{3}{2}}\\)",
"\\dfrac{4}{3} \\pi \\left( \\dfrac{r \\sqrt{l^2 - r^2}}{l + r} \\right)^3"
] | [
"A cone with radius \\( r \\) and lateral edge \\( l \\) is filled with water. The largest possible iron sphere is lowered into the cone so that no part of it is outside the liquid. What volume of water overflows?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 14,
"tir": 6
} | 47 | 0 | 0 |
omr-ws-fe2661f1709606f026a5ce1276f7ecc3681ec2565cc69d9be03e7bb87ceac459 | A cone with the altitude $H$ and the angle between the generator and altitude equal to $\alpha $ is cut by a spherical surface with the center at the vertex of the cone to divide the volume of the cone into two equal portions. Find the radius of the sphere. | [
"$r=\\frac{H}{2}\\sqrt [3]{\\frac{\\tan^2 \\alpha }{\\sin^2 \\frac{\\alpha }{2}}}$"
] | [
"A cone with the altitude $H$ and the angle between the generator and altitude equal to $\\alpha $ is cut by a spherical surface with the center at the vertex of the cone to divide the volume of the cone into two equal portions. Find the radius of the sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bdc1167cd2649a75b38e22a77a7293bc72a743db2ff5d10f3b1694b8950028b | A cone with volume $54\pi$ is cut by a plane parallel to the base, $\dfrac{1}{3}$ of the way up the height of the cone (from the base). Find the volume of the resulting frustum. | [
"38\\pi"
] | [
"A cone with volume $54\\pi$ is cut by a plane parallel to the base, $\\dfrac{1}{3}$ of the way up the height of the cone (from the base). Find the volume of the resulting frustum."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-9697ef1b3923cc0c42e4c8032e796c90aaa49f82cb6147d45c0a8e5cf5c47a0c | A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet? | [
"4000"
] | [
"A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 3
} | 35 | 0 | 0 |
omr-ws-58b91c3790da24ca88e8600be24ddc524436ae3881b0de59ae058a3d66404570 | A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top \(\frac{1}{8}\) of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet? | [
"4000"
] | [
"A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top \\(\\frac{1}{8}\\) of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d5be518fc359c1566700918ee89b806cb8d1d5d3da46f1551e53afea0846ca27 | A cone-shaped paper drinking cup is to hold 20 cm^3 of water. Find the height and radius of the cup that will require the least amount of paper. | [
"\\sqrt[3]{\\dfrac{480}{\\pi}}"
] | [
"A cone-shaped paper drinking cup is to hold 20 cm^3 of water. Find the height and radius of the cup that will require the least amount of paper."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-6ca2e10c700354491f0ad20f7a6992534ac5b01b109cc4764fae3d90f6885f99 | A confectioner makes 5 different types of pies: a, b, c, d, e. He wants to deliver pies to a festival of 120 people. He wants to fulfill the first choice of 100 people, and the remaining 20 people should not be able to get either their first or second choice. What is the least number of pies he should bring? | [
"120"
] | [
"A confectioner makes 5 different types of pies: a, b, c, d, e. He wants to deliver pies to a festival of 120 people. He wants to fulfill the first choice of 100 people, and the remaining 20 people should not be able to get either their first or second choice. What is the least number of pies he should bring?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-f8e92a8ca0aae981e427e23921d45103ca99643b5008fca258bb6d5f2076bb91 | A confectioner sells two types of nut mixtures. The standard-mixture package contains 100g of cashews and 200g of peanuts and sells for $1.95. The deluxe-mixture package contains 150g of cashews and 50g of peanuts and sells for $2.25. The confectioner has 15kg of cashews and 20kg of peanuts available. Based on past sal... | [
"40"
] | [
"A confectioner sells two types of nut mixtures. The standard-mixture package contains 100g of cashews and 200g of peanuts and sells for $1.95. The deluxe-mixture package contains 150g of cashews and 50g of peanuts and sells for $2.25. The confectioner has 15kg of cashews and 20kg of peanuts available. Based on pas... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-a70deaea7a5c3d6e88e739cb666fa48ce14e3c882a0fda4e7117b02e0d737407 | A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the $(k + 1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1 \le k \le n-1$. No chair can be occupied by more than one delegate. Find... | [
"\\( n = 2^k \\) for non-negative integers \\( k \\)."
] | [
"A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the $(k + 1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1 \\le k \\le n-1$. No chair can be occupied by more than one delegat... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 29
} | 46 | 0 | 0 |
omr-ws-2827b9db72c1aedb5a6d2124d4c4c83d083471f1d8bf215025bd868950672c92 | A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the $(k + 1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1 \leq k \leq n-1$. No chair can be occupied by more than one delegate. Fi... | [
"\\( n = 2^k \\) for \\( k \\in \\mathbb{N} \\)"
] | [
"A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the $(k + 1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1 \\leq k \\leq n-1$. No chair can be occupied by more than one deleg... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-0118b4890d4bb15ad9a9d75a22b143ac3ec0c5cc1e021d4e0fd6903428f7d3ce | A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the $(k+1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1 \leq k \leq n-1$. No chair may be occupied by more than one delegate. Find... | [
"\\( n = 2^k \\) for \\( k \\geq 0 \\)"
] | [
"A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the $(k+1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1 \\leq k \\leq n-1$. No chair may be occupied by more than one delegat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-360b568b95fa57f73ade6e2be49d2c9661f67e2e6cd55c37c45a52611ad9a82b | A conference hall has a round table with \( n \) chairs. There are \( n \) delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the \( (k+1) \)th delegate sits \( k \) places to the right of the \( k \)th delegate, for \( 1 \le k \le n-1 \). (In particular, the second delegat... | [
"\\( n = 2^k \\) for non-negative integers \\( k \\)"
] | [
"A conference hall has a round table with \\( n \\) chairs. There are \\( n \\) delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the \\( (k+1) \\)th delegate sits \\( k \\) places to the right of the \\( k \\)th delegate, for \\( 1 \\le k \\le n-1 \\). (In particular,... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c9b36707021cc25a6b5e8d7cfa1193ada76a2c13851b44412b71c4f771b96ce9 | A conference hall has a round table with \(n\) chairs. There are \(n\) delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the \((k+1)\)-th delegate sits \(k\) places to the right of the \(k\)-th delegate, for \(1 \leq k \leq n-1\). (In particular, the second delegate sits n... | [
"\\( n = 2^k \\) for \\( k \\in \\mathbb{N} \\) (i.e., \\( n \\) is a power of 2)."
] | [
"A conference hall has a round table with \\(n\\) chairs. There are \\(n\\) delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter, the \\((k+1)\\)-th delegate sits \\(k\\) places to the right of the \\(k\\)-th delegate, for \\(1 \\leq k \\leq n-1\\). (In particular, the sec... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-83d1b2a5b379ad6d68be45b2bfa3dae2812458ea373eca55a2c71ac54f057ecd | A conference has 47 people attending. One woman knows 16 of the men who are attending, another knows 17, and so on up to the last woman who knows all the men who are attending. Find the number of men and women attending the conference. | [
"16 women, 31 men."
] | [
"A conference has 47 people attending. One woman knows 16 of the men who are attending, another knows 17, and so on up to the last woman who knows all the men who are attending. Find the number of men and women attending the conference."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 30
} | 63 | 0 | 0 |
omr-ws-2e48b19e414cdba55e17e0f4d2832631cc15081f903205b4972e751178e5785e | A conference has 47 people attending. One woman knows 16 of the men, another knows 17, and so on up to the $m$-th woman who knows all $n$ of the men attending. Find $m$ and $n$. | [
"\\( m = 16 \\), \\( n = 31 \\)"
] | [
"A conference has 47 people attending. One woman knows 16 of the men, another knows 17, and so on up to the $m$-th woman who knows all $n$ of the men attending. Find $m$ and $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 12
} | 17 | 0 | 0 |
omr-ws-ac922616bad199e8abfd765b3019ca3c2be90ff962eb97a3578b31020d0d1861 | A configuration is a finite set \( S \) of points in the plane such that no three points are collinear, and each point is assigned a color. If a triangle formed by vertices in \( S \) has one angle measuring at least 120°, then exactly two of its vertices must be of the same color. Find the maximum number of points tha... | [
"6"
] | [
"A configuration is a finite set \\( S \\) of points in the plane such that no three points are collinear, and each point is assigned a color. If a triangle formed by vertices in \\( S \\) has one angle measuring at least 120°, then exactly two of its vertices must be of the same color. Find the maximum number of p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-41fde426de0e8e181523d03297ff4873374c10cb9f1b0e8ec0780de251d200a3 | A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points are collinear. An arrangement of lines is good for a Colombian configuration if the following conditions are satisfied: - No line passes through any point of the confi... | [
"2013"
] | [
"A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points are collinear. An arrangement of lines is good for a Colombian configuration if the following conditions are satisfied:\n- No line passes through any point of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-04a81f4405a5a7af06bf9a7a03f61cc64f2cc2a8afeade53310fe75c14ccb25e | A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two con... | [
"2013"
] | [
"A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following tw... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e17974d1ce27acc0ba7c5b1064d1d48a7164a15c8461598b70393be4272c481a | A congruum is defined as the difference between two legs of a right triangle with integer sides. Given that the equation $x^4 - y^4 = z^2$ has no nonzero integer solutions, find the sum of all possible values of $k$ such that $k^2$ cannot be a congruum. | [
"no_valid_answer_found"
] | [
"A congruum is defined as the difference between two legs of a right triangle with integer sides. Given that the equation $x^4 - y^4 = z^2$ has no nonzero integer solutions, find the sum of all possible values of $k$ such that $k^2$ cannot be a congruum."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26d10544d45b001a5b08f1c836bc1f668605e0ab30d89a92bc640ddb0b3d6f78 | A conic $\omega$ touches the sides $BC$, $CA$, and $AB$ of $\triangle ABC$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $X$ be a point on the projective plane (possibly a point at infinity) that projects $A_1$, $B_1$, and $C_1$ on $\omega$ to points $A_2$, $B_2$, and $C_2$, respectively. If the coordinates of $... | [
"(1,1)"
] | [
"A conic $\\omega$ touches the sides $BC$, $CA$, and $AB$ of $\\triangle ABC$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $X$ be a point on the projective plane (possibly a point at infinity) that projects $A_1$, $B_1$, and $C_1$ on $\\omega$ to points $A_2$, $B_2$, and $C_2$, respectively. If the coordina... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-55bd5e3576c59850528667672649821c49ce6bba580746c0ae1aa03cb7fc63f3 | A conic intersects the sidelines of triangle \(ABC\) at points \(A_1, A_2 \in BC\), \(B_1, B_2 \in CA\), and \(C_1, C_2 \in AB\). Define the following points: - \(A_3 = AA_2 \cap B_1C_1\), \(B_3 = BB_2 \cap C_1A_1\), \(C_3 = CC_2 \cap A_1B_1\), - \(A_4 = AA_1 \cap B_2C_2\), \(B_4 = BB_1 \cap C_2A_2\), \(C_4 = CC_1 \cap... | [
"1"
] | [
"A conic intersects the sidelines of triangle \\(ABC\\) at points \\(A_1, A_2 \\in BC\\), \\(B_1, B_2 \\in CA\\), and \\(C_1, C_2 \\in AB\\). Define the following points:\n- \\(A_3 = AA_2 \\cap B_1C_1\\), \\(B_3 = BB_2 \\cap C_1A_1\\), \\(C_3 = CC_2 \\cap A_1B_1\\),\n- \\(A_4 = AA_1 \\cap B_2C_2\\), \\(B_4 = BB_1 \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8c7b7c7088b4d2440de85c349043f55b54d631634355e9fe7004201bd032616 | A conic with apothem 1 varies in height and radius (with \( r < \frac{1}{2} \)) such that the area of the conic is 9 times the area of its inscribed sphere. What is the height of the conic? | [
"\\dfrac{\\sqrt{828 + 204\\sqrt{7}}}{37}"
] | [
"A conic with apothem 1 varies in height and radius (with \\( r < \\frac{1}{2} \\)) such that the area of the conic is 9 times the area of its inscribed sphere. What is the height of the conic?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-3e1e6458df519736c245b541dc698292f37301920764e1e5de9f70d9d9361ce1 | A conical bottle perched on its base is filled with water up to a height that is 8 cm from its vertex. When the bottle is turned over, the water level is at 2 cm from its base. Calculate the height of the bottle. | [
"10"
] | [
"A conical bottle perched on its base is filled with water up to a height that is 8 cm from its vertex. When the bottle is turned over, the water level is at 2 cm from its base. Calculate the height of the bottle."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-0af1dadfdcc79cd7750498e5241d519d953495f39798787ed1f8571d04072048 | A conical glass is in the form of a right circular cone with a slant height of 21 and a radius of the top rim of 14. An ant is at the midpoint of a slant on the outside of the glass and sees a honey drop diametrically opposite to it on the inside wall of the glass. If \( d \) is the shortest distance the ant should cra... | [
"36"
] | [
"A conical glass is in the form of a right circular cone with a slant height of 21 and a radius of the top rim of 14. An ant is at the midpoint of a slant on the outside of the glass and sees a honey drop diametrically opposite to it on the inside wall of the glass. If \\( d \\) is the shortest distance the ant sho... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95b42b20271698f84007eb0bc3f976cd32e8113496b50a88902d7c160c572d62 | A conical hat is 40 cm tall and has a perimeter along the brim of 60π cm. What is the radius of the largest sphere that can be put on a table and covered entirely by the hat, so that the hat rests on the table surface and conceals the sphere? | [
"15"
] | [
"A conical hat is 40 cm tall and has a perimeter along the brim of 60π cm. What is the radius of the largest sphere that can be put on a table and covered entirely by the hat, so that the hat rests on the table surface and conceals the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47405bced935e02cb345472fc0024a833f53481b5e3c3cf857a66134de2cdeaa | A conical paper cup has a depth $h$ and a radius of the rim $r$. The expression $\frac{3\sqrt{r^2 + h^2}}{r}$ represents the amount of paper used per unit volume. If the depth $h$ is $\sqrt{2}$ times the radius $r$, find the simplified form of the expression $\frac{3\sqrt{r^2 + h^2}}{r}$ and express your answer as $a\s... | [
"6"
] | [
"A conical paper cup has a depth $h$ and a radius of the rim $r$. The expression $\\frac{3\\sqrt{r^2 + h^2}}{r}$ represents the amount of paper used per unit volume. If the depth $h$ is $\\sqrt{2}$ times the radius $r$, find the simplified form of the expression $\\frac{3\\sqrt{r^2 + h^2}}{r}$ and express your answ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-feabb89f1d50309496a4ebe15cb3168928cefa2506838341560ee21a566ec7d2 | A conical paper cup has a diameter of 8 cm and a depth of 10 cm. Root beer leaves the cup at a rate of 7 cm³/sec. At what rate is the level of the liquid in the cup changing when the liquid is 6 cm deep? | [
"\\(- \\frac{175}{144 \\pi}\\)"
] | [
"A conical paper cup has a diameter of 8 cm and a depth of 10 cm. Root beer leaves the cup at a rate of 7 cm³/sec. At what rate is the level of the liquid in the cup changing when the liquid is 6 cm deep?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-25ddeff5fda2c39e952a37b50f1f2f7e3d2ad2e1300e47519ee00ed37501c0d9 | A conical reservoir with a diameter of 30 ft and a depth of 10 ft is filled to the top with water of density 62.4 lb/cubic foot. Calculate the work done by pumping all the water over the top of the reservoir. | [
"367566 lb*ft."
] | [
"A conical reservoir with a diameter of 30 ft and a depth of 10 ft is filled to the top with water of density 62.4 lb/cubic foot. Calculate the work done by pumping all the water over the top of the reservoir."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-b5b18e104bc07e1dc23761b29bdb00dbb3ef4fb4a637668b5d4a0f3ae52b62ae | A conical tank has a radius of 10 feet and is 24 feet high. If water flows in at 20 ft³/min, how fast is the depth of the water changing when the water is 6 feet deep, assuming the cone is lying on its side? | [
"\\dfrac{16}{5\\pi}"
] | [
"A conical tank has a radius of 10 feet and is 24 feet high. If water flows in at 20 ft³/min, how fast is the depth of the water changing when the water is 6 feet deep, assuming the cone is lying on its side?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bf6d8624f6d70333009e13b7441f7edad92aa251867fa33347d7d11eda4f4afa | A conical tank with height 12 feet and radius 4 feet is filled with liquid. How much liquid has poured from the tip of the cone if the water level is 9 feet from the tip? | [
"\\( 37\\pi \\)"
] | [
"A conical tank with height 12 feet and radius 4 feet is filled with liquid. How much liquid has poured from the tip of the cone if the water level is 9 feet from the tip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-0f87c57b99e513f7d40bd835e94798079b03551d07832640868ddee7a9f37767 | A conical tank with radius 4 and height 12 is filled with liquid. How much liquid has poured from the tip of the cone if the water level is 9 feet from the tip? | [
"\\( 37\\pi \\)"
] | [
"A conical tank with radius 4 and height 12 is filled with liquid. How much liquid has poured from the tip of the cone if the water level is 9 feet from the tip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-666b67da5196cdb29b1b8f04589a9fd0b0b6c7fbc3c3d8d849d56852d4a2ec16 | A conical tent is made using a semi-circular piece of canvas with a radius of 8 feet. Find the height of the tent and the volume of air inside the tent in cubic feet. | [
"Height = \\( 4\\sqrt{3} \\) feet, Volume = \\( \\frac{64\\pi\\sqrt{3}}{3} \\) cubic feet"
] | [
"A conical tent is made using a semi-circular piece of canvas with a radius of 8 feet. Find the height of the tent and the volume of air inside the tent in cubic feet."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 25
} | 30 | 0 | 0 |
omr-ws-3b327dca7af6ea5dc010fdf31df77bf4e73c5de1d8d071a9d4f23a29442cf17c | A conical vessel with radius \( r = 12 \) cm and depth \( h = 16 \) cm is completely filled with water. A sphere is lowered into the water such that it touches the inner curved surface of the vessel and is just immersed up to its topmost point. What is the quantity of water that flows out when the sphere is lowered int... | [
"\\( 288\\pi \\)"
] | [
"A conical vessel with radius \\( r = 12 \\) cm and depth \\( h = 16 \\) cm is completely filled with water. A sphere is lowered into the water such that it touches the inner curved surface of the vessel and is just immersed up to its topmost point. What is the quantity of water that flows out when the sphere is lo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-c2f167e230703c897255f2abf666a493b115aa32708ef50b1f757b685c984bce | A conman bought a $10$ dollar magazine from a store with a counterfeit $100$ dollar bill. The store owner did not have change, so they went to a nearby cafe and exchanged the counterfeit bill for $100$ dollars in genuine currency. The store owner then returned $90$ dollars to the conman. Later, the cafe owner discovere... | [
"100 dollars"
] | [
"A conman bought a $10$ dollar magazine from a store with a counterfeit $100$ dollar bill. The store owner did not have change, so they went to a nearby cafe and exchanged the counterfeit bill for $100$ dollars in genuine currency. The store owner then returned $90$ dollars to the conman. Later, the cafe owner disc... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b279b22800302c6cb9cce0c0d15c4213a8bd9567be8d7f8f6b3d6a08ab0391b3 | A connected tree \( T \) has 500 vertices, each with degree 1, 2, or 3. Consider a black-and-white coloring of these vertices called [b]interesting[/b] if more than half of the vertices are white but no two white vertices are connected. Let \( S \) be a subset of vertices such that in any [b]interesting[/b] coloring, m... | [
"250"
] | [
"A connected tree \\( T \\) has 500 vertices, each with degree 1, 2, or 3. Consider a black-and-white coloring of these vertices called [b]interesting[/b] if more than half of the vertices are white but no two white vertices are connected. Let \\( S \\) be a subset of vertices such that in any [b]interesting[/b] co... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-a02604b5aac01e54d06d310b96d0a72e3c044077f55012cff08fc0a73c8d9353 | A construction crew has realized that the remaining work before completion will require \( n \) single-day projects and \( m \) two-day projects. A half of a two-day project can be completed and the remainder left for any other day (meaning that the completion of a two-day project becomes a single-day project in the re... | [
"2m + \\frac{n m}{m + 1} - H_m"
] | [
"A construction crew has realized that the remaining work before completion will require \\( n \\) single-day projects and \\( m \\) two-day projects. A half of a two-day project can be completed and the remainder left for any other day (meaning that the completion of a two-day project becomes a single-day project ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17eb63cfcacfdb71fe965c8dc3b23a931418eb0ff42a67a0fde5eebf3f671d7b | A construction job required the digging out of 8000 ${\rm m}^3$ of earth in a specified time. The operation was completed 8 days ahead of time because the team of navvies overfulfilled their plan by 50 cubic meters daily. Determine the original time limit for the assignment and daily overfulfillment of the plan in perc... | [
"The original time limit was 40 days, the daily overfulfilment of the plan being 25 per cent"
] | [
"A construction job required the digging out of 8000 ${\\rm m}^3$ of earth in a specified time. The operation was completed 8 days ahead of time because the team of navvies overfulfilled their plan by 50 cubic meters daily. Determine the original time limit for the assignment and daily overfulfillment of the plan i... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bddd9f4bae1508e3289e44b2c83d8f116175948152c9c5ae3291adc0a8b57a53 | A consultant unpacks six laptop computers, uses them to conduct a training session, and then repackages them for shipment back to her company. After repackaging the last computer, she notices that each box has on it a serial number that should be the same as the number on the machine within. The repackaging was conduct... | [
"\\(\\frac{191}{720}\\)"
] | [
"A consultant unpacks six laptop computers, uses them to conduct a training session, and then repackages them for shipment back to her company. After repackaging the last computer, she notices that each box has on it a serial number that should be the same as the number on the machine within. The repackaging was co... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1faf9b528e6678edbfa22c825234720522d507c9b29965aa6ecb4b246ce8f61 | A consumer purchases 12 frozen chickens, and it is estimated that 30% of frozen chickens contain enough salmonella bacteria to cause illness if improperly cooked. What is the probability that the consumer will have more than 6 contaminated chickens? | [
"0.118"
] | [
"A consumer purchases 12 frozen chickens, and it is estimated that 30% of frozen chickens contain enough salmonella bacteria to cause illness if improperly cooked. What is the probability that the consumer will have more than 6 contaminated chickens?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-479eddb81779484a02b2eec17ec711da8196ff288fecaced8d46529f94096679 | A consumer spends an amount \( m \) to buy \( x \) units of one good at the price of 6 per unit and \( y \) units of a different good at a price of 10 per unit. The consumer's utility function is \( U(x, y) = xy + y^2 + 2x + 2y \). The problem is to maximize \( U(x, y) \) subject to the constraint \( 6x + 10y = m \), w... | [
"\\( x^* = \\frac{40 - m}{24}, y^* = \\frac{m - 8}{8}, \\lambda = \\frac{m + 8}{48} \\)"
] | [
"A consumer spends an amount \\( m \\) to buy \\( x \\) units of one good at the price of 6 per unit and \\( y \\) units of a different good at a price of 10 per unit. The consumer's utility function is \\( U(x, y) = xy + y^2 + 2x + 2y \\). The problem is to maximize \\( U(x, y) \\) subject to the constraint \\( 6x... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 12
} | 17 | 0 | 0 |
omr-ws-e46ec32611b494b94b8db71b9145b5ab9e77ac45f7ef78ba478d98199febca44 | A container A contains 6 red and 4 green spheres, and a container B contains 7 red and 3 green spheres. A sphere is randomly chosen from container A and placed into container B. Then, a sphere is randomly chosen from container B and placed back into container A. Find the probability of the following events: (a) A red s... | [
"\\(\\frac{24}{55}\\) for event (a) and \\(\\frac{32}{55}\\) for event (b)"
] | [
"A container A contains 6 red and 4 green spheres, and a container B contains 7 red and 3 green spheres. A sphere is randomly chosen from container A and placed into container B. Then, a sphere is randomly chosen from container B and placed back into container A. Find the probability of the following events:\n(a) A... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-5ebb9c28fe6cd44d2af8f0fa4129fa77b15ce97948bde25bec5fd60d5771c2a7 | A container consists of a right circular cone with a base radius of 2 cm and height of 6 cm, surmounted by a frustum with bases of radii 2 cm and 8 cm and height of 6 cm. If there is a rainfall measuring 2 cm, the rain falling into the container will fill it to a height of \( m + 3\sqrt{n} \) cm, where \( m \) and \( n... | [
"9"
] | [
"A container consists of a right circular cone with a base radius of 2 cm and height of 6 cm, surmounted by a frustum with bases of radii 2 cm and 8 cm and height of 6 cm. If there is a rainfall measuring 2 cm, the rain falling into the container will fill it to a height of \\( m + 3\\sqrt{n} \\) cm, where \\( m \\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7a950f3abe99f3f919c0ae2508ef89d458c3a8b37c54705ac31e9df65619be3 | A container has 4 quarts of maple syrup. One quart is removed and replaced with one quart of corn syrup, and the result is thoroughly mixed. This operation is repeated three more times. How much maple syrup has been taken out by these four removals? | [
"2 47/64"
] | [
"A container has 4 quarts of maple syrup. One quart is removed and replaced with one quart of corn syrup, and the result is thoroughly mixed. This operation is repeated three more times. How much maple syrup has been taken out by these four removals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-3025a9f5ac7d87d67daa7e517c5c11b2f7ff1ae66668c0cf7d7f1f9a0ea77c23 | A container holds \(42\) orange marbles, \(25\) purple marbles, \(30\) pink marbles, \(17\) teal marbles, \(22\) brown marbles, and \(15\) gray marbles. What is the minimum number of marbles that must be drawn from the container without replacement to ensure that at least \(18\) marbles of a single color will be drawn? | [
"101"
] | [
"A container holds \\(42\\) orange marbles, \\(25\\) purple marbles, \\(30\\) pink marbles, \\(17\\) teal marbles, \\(22\\) brown marbles, and \\(15\\) gray marbles. What is the minimum number of marbles that must be drawn from the container without replacement to ensure that at least \\(18\\) marbles of a single c... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 12
} | 60 | 0 | 0 |
omr-ws-ffbaaef64baf7a5a95a384e5cc5eff0dd09e0be268b5309cb54d6a38c39643e2 | A container in the shape of a parallelepiped has dimensions that are distinct positive integers. The container is filled with two unmixed liquids, $A$ and $B$, where the volume of $A$ is $36\%$ of the volume of $B$. When the container is supported horizontally on any of its six faces, the height that each of the two li... | [
"340"
] | [
"A container in the shape of a parallelepiped has dimensions that are distinct positive integers. The container is filled with two unmixed liquids, $A$ and $B$, where the volume of $A$ is $36\\%$ of the volume of $B$. When the container is supported horizontally on any of its six faces, the height that each of the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dc8f4578987fec2bc0d43376b1a1aa5036b67b17d48fb20733d908d5fdb447f | A container in the shape of a pyramid has a 12 × 12 square base, and the other four edges each have length 11. The container is partially filled with liquid so that when one of its triangular faces is lying on a flat surface, the level of the liquid is half the distance from the surface to the top edge of the container... | [
"231"
] | [
"A container in the shape of a pyramid has a 12 × 12 square base, and the other four edges each have length 11. The container is partially filled with liquid so that when one of its triangular faces is lying on a flat surface, the level of the liquid is half the distance from the surface to the top edge of the cont... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-3dcb87c1aa089b6e49ca5082d80c685a743d64931bdad8f59467c4e6b8810d2b | A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$ -inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is... | [
"052"
] | [
"A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$ -inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the li... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 3,
"tir": 12
} | 49 | 0 | 0 |
omr-ws-3f06ee16ef3c0d6413b7b233330381125fafec093e2b7f86fa2f9ab93c9f9db7 | A container in the shape of a right circular cone is 12 inches tall with a base radius of 5 inches. When the cone is held with its point down and its base horizontal, the liquid inside is 9 inches deep. When the cone is held with its point up and its base horizontal, the depth of the liquid is expressed as $m - n\sqrt[... | [
"052"
] | [
"A container in the shape of a right circular cone is 12 inches tall with a base radius of 5 inches. When the cone is held with its point down and its base horizontal, the liquid inside is 9 inches deep. When the cone is held with its point up and its base horizontal, the depth of the liquid is expressed as $m - n\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-cc40e30b2be0ddcbb782b50e3b9afeb73e5c8fdb22c7986d9d21f89b79de199a | A container is shaped like a right circular cone with a base diameter of 18 and height of 12. Four spheres, each with radius 3, are placed inside the container such that the first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are each tangent to the cone and ... | [
"704"
] | [
"A container is shaped like a right circular cone with a base diameter of 18 and height of 12. Four spheres, each with radius 3, are placed inside the container such that the first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are each tangent to the cone... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
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